Monday, 24 February 2014

Sunday Afternoon Maths I Answers & Extensions

Burning Ropes

Light one rope at both ends and the other at one end. When the first rope finishes burning, light the other end of the second rope. The second rope will finish burning 45 minutes after the start.

Extension

What are all the possible times you can measure with two ropes? How about three ropes? Four ropes? n ropes?

Pole Position

The two poles can be any distance apart; the distance does not affect the heights.

Extension

If the heights of the two poles were 12m and 24m tall, what height would the intersection of the lines be?
If the heights of the two poles were am and bm tall, what height would the intersection of the lines be?

Circles

Extension

Prove that the red and blue areas are equal.

Pyramid and Tetrahedron

Let the length of a side of a triangle be 2L(I used 2L and not L to get rid of fractions in the calculations).
By Pythagoras' Theorem, the height of a triangle is L√3.
Using Pythagoras' Theorem again, the height of the square-based pyramid is L√2.
Therefore, the volume of the square-based pyramid is ⅓×(2L)2×L√2.
This simplifies to 4/3√2L3.

Next, we find the area of the tetrahedron.
Call the point on the base of the tetrahedron, directly below the vertex at the top A.
Using cosine in the triangle made by A, the corner of the base and the midpoint of a side of the base, the distance from the corner to A is 2/√3L.
Using Pythagoras' Theorem yet again, we find that the height of the tetrahedron is L2√2/√3
Therefore, the volume of the tetrahedron is ⅓×½×2L×L√3×L2√2/√3.
This simplifies to 2/3√2L3.

Finally, the ratio of volume of the square based pyramid to the tetrahedron is:4/3√2L3:2/3√2L3
2 : 1